In cities, you'll have a few 'Liberation Contract' icons on your map. The first time you meet these guys, they'll tell you how shitty conditions are, and that they like what you're trying to do. They might even give you specific tasks: like a doctor in New York telling you to get rid of blankets infected with smallpox. You'll also have several 'Liberation Missions' on your map. These range from defending a farmer as he harvests his land to putting down rabid dogs or the aformentioned burning of infected blankets. Each one you do (I think there's three of each type) adds up to your total control of that particular district. Not every mission will appear on your map, so just keep an eye out while you're exploring the cities if you're still short a few. You can also check your progress in the Log section of the map, and see what kinds of missions you still need to do.
Once a district has been liberated, head back to your contact, who will be marked on the map with a black 'Liberation Mission'-icon. In a cutscene, your contact will point you in the way of the chief Templar overseeing the district, and volunteer to help kill him. During that mission, you'll also unlock that Recruit's specific ability: for instance, starting a riot or setting up an ambush. Once that mission is complete, your contact will swear allegiance to you and you'll have a new Assassin trainee.
Each trainee starts at level 0, and maxes out at level 11. They'll gain XP by coming to your aid in the overworld, but you can also send them on missions like in the previous games. Hold down the RB button (on the Xbox) to open up the Assassin menu. You can select one of your recruit's abilities from there, but if you press X you open the map of the Colonies. Each colony has a few missions tied to it. You'll need to build up your Assassins' skills before you can send them on the high-risk missions: this takes a longer time than it did in the previous games. XP is also earned a lot slower, and once you gain control of a Colony, the Templar's can't take it over again like they could in Revelations. At least, not to my knowledge.
Completing missions brings in coin and the occasional item, alongside XP for your Recruit. At higher levels, they're stronger and more resilient and can come to your aid more often. Assuming they're not on an adventure in the Colonies, that is.
Hope that helps for everyone having trouble with the system.
This is a great guide man. Was that so hard Ubi?
I'm saving this post for later use.